Find the Cartesian equations of the line which passes through the point
Question: Find the Cartesian equations of the line which passes through the point $(-2,4,-5)$ and which is parallel to the line $\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}$. Solution:...
Read More →Write the equations of a line parallel to the line
Question: Write the equations of a line parallel to the line $\frac{x-2}{-3}=\frac{y+3}{2}=\frac{z+5}{6}$ and passing through the point $(1,-2,3)$. Solution: is given by...
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Question: If the equations of a line are $\frac{3-x}{-3}=\frac{y+2}{-2}=\frac{z+2}{6}$ find the direction cosines of a line parallel to the given line. Solution:...
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Question: Find the direction cosines of the line $! \frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$. Solution:...
Read More →If a line has direction ratios
Question: If a line has direction ratios $2,-1,-2$ then what are its direction cosines? Solution:...
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Question: Show that the lines $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5}$ and $\frac{x-2}{2}=\frac{y-1}{3}=\frac{z+1}{-2}$ do not intersect each other. Solution:...
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Question: Show that the lines $\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$ and $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ intersect and find their point of intersection. Solution:...
Read More →Find the length and the equations of the line of shortest distance between the lines given by:
Question: Find the length and the equations of the line of shortest distance between the lines given by: $\frac{x-6}{3}=\frac{y-7}{-1}=\frac{z-4}{1}$ and $\frac{x}{-3}=\frac{y+9}{2}=\frac{z-2}{4}$ Solution:...
Read More →Find the length and the equations of the line of shortest distance between the lines given by:
Question: Find the length and the equations of the line of shortest distance between the lines given by: $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ and $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$. Solution:...
Read More →Find the length and the equations of the line of shortest distance between the lines given by:
Question: Find the length and the equations of the line of shortest distance between the lines given by: $\frac{x-3}{-1}=\frac{y-4}{2}=\frac{z+2}{1}$ and $\frac{x-1}{1}=\frac{y+7}{3}=\frac{z+2}{2}$. Solution:...
Read More →Find the length and the equations of the line of shortest distance between
Question: Find the length and the equations of the line of shortest distance between the lines given by: $\frac{x-3}{3}=\frac{y-8}{-1}=z-3$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4} .$ Solution:...
Read More →Find the shortest distance between the lines given below:
Question: Find the shortest distance between the lines given below: $\frac{x-12}{-9}=\frac{y-1}{4}=\frac{z-5}{2}$ and $\frac{x-23}{-6}=\frac{y-10}{-4}=\frac{z-25}{3}$ HINT: Change the given equations in vector form. Solution:...
Read More →Find the shortest distance between the lines given below:
Question: Find the shortest distance between the lines given below: $\frac{x-1}{-1}=\frac{y+2}{1}=\frac{z-3}{-2}$ and $\frac{x-1}{2}=\frac{y+1}{2} \frac{z+1}{-2}$. Solution:...
Read More →Write the vector equation of the following lines and hence find the shortest distance between them :
Question: Write the vector equation of the following lines and hence find the shortest distance between them : $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-5}{5}$. Solution: ....
Read More →Write the vector equation of each of the following lines and hence determine the distance between them:
Question: Write the vector equation of each of the following lines and hence determine the distance between them: $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $\frac{x-3}{4}=\frac{y-3}{6}=\frac{z+5}{12}$ HINT: The given lines are $\mathrm{L}_{1}: \overrightarrow{\mathrm{r}}=(-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}})+\lambda(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ $\mathrm{L}_{2}: \overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})+2 \mu(2 \h...
Read More →Find the vector equation of a line passing through the point
Question: Find the vector equation of a line passing through the point $(2,3,2)$ and parallel to the line $\vec{r}=(-2 \hat{i}+3 \hat{j})+\lambda(2 \hat{i}-3 \hat{j}+6 \hat{k})$ Also, find the distance between these lines. HINT: The given line is $L_{1}: \vec{r}=(-2 \hat{i}+3 \hat{j})+\lambda(2 \hat{i}-3 \hat{j}+6 \hat{k})$ The required line is $\mathrm{L}_{2}: \overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+6 \hat{...
Read More →Find the distance between the parallel lines
Question: Find the distance between the parallel lines $L_{1}$ and $L_{2}$ whose vector equations are $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$, and $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ Solution:...
Read More →Find the shortest distance between the lines
Question: Find the shortest distance between the lines $L_{1}$ and $L_{2}$ whose vector equations are $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ HINT: The given lines are parallel. Solution:...
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Question: Show that the lines $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(4 \hat{\mathrm{i}}+\hat{\mathrm{j}})+\mu(5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ intersect. Also, find their point of intersection. Solution:...
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Question: Show that the lines $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+2 \mathrm{j}+3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\mu$ $(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$ intersect. Also, find their point of intersection. Solution:...
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Question: Show that the lines $\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}-15 \hat{\mathrm{j}}+9 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+9 \hat{\mathrm{k}})+\mu$ $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ do not intersect. Solution:...
Read More →Compute the shortest distance between the lines
Question: Compute the shortest distance between the lines $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\lambda(2 \hat{\mathrm{i}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}})+\mu(\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})$ Determine whether these lines intersect or not. Solution:...
Read More →Find the shortest distance between the given lines.
Question: Find the shortest distance between the given lines. $\overrightarrow{\mathrm{r}}=(\lambda-1) \hat{\mathrm{i}}+(\lambda+1) \hat{\mathrm{j}}-(\lambda+1) \hat{\mathrm{k}}$ $\overrightarrow{\mathrm{r}}=(1-\mu) \hat{\mathrm{i}}+(2 \mu-1) \hat{\mathrm{j}}+(\mu+2) \hat{\mathrm{k}}$ Solution:...
Read More →Find the shortest distance between the given lines.
Question: Find the shortest distance between the given lines. $\overrightarrow{\mathrm{r}}=(3-\mathrm{t}) \hat{\mathrm{i}}+(4+2 \mathrm{t}) \hat{\mathrm{j}}+(\mathrm{t}-2) \hat{\mathrm{k}}$ $\overrightarrow{\mathrm{r}}=(1+s) \hat{\mathrm{i}}+(3 \mathrm{~s}-7) \hat{\mathrm{j}}+(2 \mathrm{~s}-2) \hat{\mathrm{k}}$ Solution:...
Read More →Find the shortest distance between the given lines.
Question: Find the shortest distance between the given lines. $\vec{r}=(6 \hat{i}+3 \hat{k})+\lambda(2 \hat{i}-\hat{j}+4 \hat{k})$ $\vec{r}=(-9 \hat{i}+\hat{j}-10 \hat{k})+\mu(4 \hat{i}+\hat{j}+6 \hat{k})$ Solution:...
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